A body of mass `m` moving with a velocity `v` in the `x` direction collides with another body of mass `M` moving in `y` direction with a velocity `V`. They coalesce into one body during collision.

Options

A. The magnitude of momentum of the composition body `[(m v)^(2) + (MV)^(2)]^(1//2)`
B. The fraction of initial `K.E.` transformed into heat is `= ((mM)/(m+M))((v^(2)+V^(2))/(mv^(2)+MV^(2)))`
C. Decrease in kinetic energy is `(mM)/(2(m+M))(v^(2)+V^(2))`
D. None of these

Answer

A, B, C

Explanation

(a) `mv = (M+m)V' cos theta`
(b) `Mv = (m+M)V' sin theta`
`(m + M)V' = sqrt((mv)^(2) + (MV)^(2))`
(b) Initial kinetic is , `k_(l) = 1/2 mv^(2) + 1/2 MV^(2)`
Final K.E. is `k_(f) = 1/2 (m + M) v'^(2)`
Decrease in K.E. `= k_(1) - k_(f) , Deltak = (mM)/(2(m+M)) (v^(2) + V^(2))`
Fraction of initial kinetic transformed into heat is
`(Deltak)/(k_(l)) = (mM)/(m+M) ((v^(2) + V^(2))/(mv^(2) + MV^(2)))`